By Chris Fenaroli, Delaney Mackenzie, and Maxwell Margenot
https://www.quantopian.com/lectures/introduction-to-pairs-trading
These exercises correspond to the Introduction to Pairs Trading lecture, which is part of the Quantopian Lecture Series. It is meant to cement a few concepts presented in the lecture, specifically cointegration and construction of cointegrated pair spreads.
We expect you to rely heavily on the code presented in the corresponding lecture. Please copy and paste regularly from that lecture when starting to work on the problems, as trying to do them from scratch will likely be too difficult. There is an answer key available for this problem set. Please email us at feedback@quantopian.com for results.
Part of the Quantopian Lecture Series:
In [2]:
# Useful Functions
def find_cointegrated_pairs(data):
n = data.shape[1]
score_matrix = np.zeros((n, n))
pvalue_matrix = np.ones((n, n))
keys = data.keys()
pairs = []
for i in range(n):
for j in range(i+1, n):
S1 = data[keys[i]]
S2 = data[keys[j]]
result = coint(S1, S2)
score = result[0]
pvalue = result[1]
score_matrix[i, j] = score
pvalue_matrix[i, j] = pvalue
if pvalue < 0.05:
pairs.append((keys[i], keys[j]))
return score_matrix, pvalue_matrix, pairs
In [1]:
# Useful Libraries
import numpy as np
import pandas as pd
import statsmodels
import statsmodels.api as sm
from statsmodels.tsa.stattools import coint, adfuller
# just set the seed for the random number generator
np.random.seed(107)
import matplotlib.pyplot as plt
We'll use some artificially generated series first as they are much cleaner and easier to work with. In general when learning or developing a new technique, use simulated data to provide a clean environment. Simulated data also allows you to control the level of noise and difficulty level for your model.
Determine whether the following two artificial series $A$ and $B$ are cointegrated using the coint() function and a reasonable confidence level.
In [3]:
A_returns = np.random.normal(0, 1, 100)
A = pd.Series(np.cumsum(A_returns), name='X') + 50
some_noise = np.random.exponential(1, 100)
B = A - 7 + some_noise
#Your code goes here
In [4]:
## answer key ##
score, pvalue, _ = coint(A,B)
confidence_level = 0.05
if pvalue < confidence_level:
print ("A and B are cointegrated")
print pvalue
else:
print ("A and B are not cointegrated")
print pvalue
A.name = "A"
B.name = "B"
pd.concat([A, B], axis=1).plot();
In [5]:
C_returns = np.random.normal(1, 1, 100)
C = pd.Series(np.cumsum(C_returns), name='X') + 100
D_returns = np.random.normal(2, 1, 100)
D = pd.Series(np.cumsum(D_returns), name='X') + 100
#Your code goes here
In [6]:
## answer key ##
score, pvalue, _ = coint(C,D)
confidence_level = 0.05
if pvalue < confidence_level:
print ("C and D are cointegrated")
print pvalue
else:
print ("C and D are not cointegrated")
print pvalue
C.name = "C"
D.name = "D"
pd.concat([C, D], axis=1).plot();
In [7]:
ual = get_pricing('UAL', fields=['price'],
start_date='2015-01-01', end_date='2016-01-01')['price']
aal = get_pricing('AAL', fields=['price'],
start_date='2015-01-01', end_date='2016-01-01')['price']
#Your code goes here
In [8]:
## answer key ##
score, pvalue, _ = coint(ual, aal)
confidence_level = 0.05
if pvalue < confidence_level:
print ("UAL and AAL are cointegrated")
print pvalue
else:
print ("UAL and AAL are not cointegrated")
print pvalue
ual.name = "UAL"
aal.name = "AAL"
pd.concat([ual, aal], axis=1).plot();
In [9]:
fcau = get_pricing('FCAU', fields=['price'],
start_date='2015-01-01', end_date='2016-01-01')['price']
hmc = get_pricing('HMC', fields=['price'],
start_date='2015-01-01', end_date='2016-01-01')['price']
#Your code goes here
In [10]:
## answer key ##
confidence_level = 0.05
score, pvalue, _ = coint(fcau, hmc)
if pvalue < confidence_level:
print ("FCAU and HMC are cointegrated")
print pvalue
else:
print ("FCAU and HMC are not cointegrated")
print pvalue
fcau.name = "FCAU"
hmc.name = "HMC"
pd.concat([fcau, hmc], axis=1).plot();
Use the find_cointegrated_pairs function, defined in the "Helper Functions" section above, to find any cointegrated pairs among a set of metal and mining securities.
Note that not all of these securities in this exercise are within the QTradableStocksUS. As you continue your development, focus on securities within the QTU to be eligible for an allocation.
In [11]:
symbol_list = ['MTRN', 'CMP', 'TRQ', 'SCCO', 'HCLP','SPY']
prices_df = get_pricing(symbol_list, fields=['price']
, start_date='2015-01-01', end_date='2016-01-01')['price']
prices_df.columns = map(lambda x: x.symbol, prices_df.columns)
#Your code goes here
In [12]:
## answer key ##
scores, pvalues, pairs = find_cointegrated_pairs(prices_df)
import seaborn
seaborn.heatmap(pvalues, xticklabels=symbol_list, yticklabels=symbol_list, cmap='RdYlGn_r'
, mask = (pvalues >= 0.99)
)
print pairs
In [13]:
S1 = prices_df['MTRN']
S2 = prices_df['SCCO']
#Your code goes here
In [14]:
## answer key ##
S1 = sm.add_constant(S1)
results = sm.OLS(S2, S1).fit()
b = results.params['MTRN']
S1 = S1['MTRN']
print b
spread = S2 - b * S1
print "p-value for in-sample stationarity: ", adfuller(spread)[1]
# The p-value is less than 0.05 so we conclude that this spread calculation is stationary in sample
spread.plot()
plt.axhline(spread.mean(), color='black')
plt.legend(['Spread']);
In [15]:
S1_out = get_pricing('MTRN', fields=['price'],
start_date='2016-01-01', end_date='2016-07-01')['price']
S2_out = get_pricing('SCCO', fields=['price'],
start_date='2016-01-01', end_date='2016-07-01')['price']
#Your code goes here
In [16]:
## answer key ##
spread = S2_out - b * S1_out
spread.plot()
plt.axhline(spread.mean(), color='black')
plt.legend(['Spread']);
print "p-value for spread stationarity: ", adfuller(spread)[1]
# Our p-value is greater than 0.05 so we conclude that this calculation of
# the spread is non-stationary out of sample
This exercise is more difficult and we will not provide initial structure.
The Hurst exponent is a statistic between 0 and 1 that provides information about how much a time series is trending or mean reverting. We want our spread time series to be mean reverting, so we can use the Hurst exponent to monitor whether our pair is going out of cointegration. Effectively as a means of process control to know when our pair is no longer good to trade.
Please find either an existing Python library that computes, or compute yourself, the Hurst exponent. Then plot it over time for the spread on the above pair of stocks.
These links may be helpful:
In [17]:
# No solution provided for extra credit exercises.
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